Dirichlet's criterion

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The criterion of Dirichlet is a mathematical convergence criterion for series . It belongs to the group of direct criteria.

Dirichlet criterion for convergence

criteria

The series

converges with if is a monotonically decreasing zero sequence and is the sequence of the partial sums

is limited .

proof

The following applies (see partial summation )

.

The first summand converges to zero, since it is bounded by a constant according to the assumption and converges to zero. The second summand even converges absolutely , because for all and with it

.

Everything is shown.

Dirichlet criterion for uniform convergence

The series

is uniformly convergent in the interval if there the partial sums of the series are uniformly bounded and if there the sequence converges uniformly to zero, for every fixed monotone.

See also

Individual evidence

  1. Harro Heuser : Textbook of Analysis . Part 1. 17th edition. Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-0777-9 , IV, sentence 33.14, p. 208/643 p .
  2. Konrad Knopp : Theory and application of the infinite series . 6th edition. Springer, Berlin / Heidelberg 1996, ISBN 3-540-59111-7 , pp. 342 ff ./604 S ( edition 1964 ( memento from January 11, 2013 in the web archive archive.today ))